An algebraic cycle relating to a section with high penetration
type: Item
instance of: scholarly article
title: An algebraic cycle relating to a section with high penetration
zbMATH Open document ID: 0756.14005
author: Takeshi Usa
published in: Journal of Mathematics of Kyoto University
publication date: 1992-06-25T00:00:00Z
review text: Let \(X\) be a smooth complex projective variety and \(E\) a holomorphic vector bundle on \(X\). If \(E\) admits a flat hermitian structure then the mapping \(\omega^ p:H^ t(X,\Omega^ s_ X\otimes E)\to H^{t+p}(X,\Omega_ X^{s+p}\otimes E)\), determined by the class \(\omega\) of a hyperplane section, is an isomorphism. The author looks at the kernel of this mapping from a point of view avoiding an explicit mention of curvature. He considers the case \(t=s=0\) --- when the elements of \(H^ t\) are sections of \(E\). He shows that if a section is transformed to the zero section and not in the kernel, then there are consequences for the algebraic cycles, provided a weakened form of the Hodge conjecture holds. The main tool is Deligne's mixed Hodge theory.
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14C30
zbMATH DE Number: 8639
zbMATH Keywords: hard Lefschetz
zbMATH Keywords: vector bundle
zbMATH Keywords: hyperplane section
zbMATH Keywords: algebraic cycles
zbMATH Keywords: Hodge conjecture
zbMATH Keywords: mixed Hodge theory
reviewed by: G. Horrocks
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250520028
OpenAlex ID: W1583851555
DOI: 10.1215/KJM/1250520028
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description: scientific article; zbMATH DE number 8639